Why is the gauge group for Ashtekar variable theory for General

As far as I understand, the gauge group for Ashtekar variable theory
for General Relativitiy is SU(2), because its group is three-
dimensional. But, why is this SU(2) instead of SO(3)? Are there any
plausible reasonings behind this? And, are there any phenomenological
differences if the gauge group were SO(3) instead of SU(2)?

Thanks in advance

Youngsub

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